Theorem isnumbasgrplem2 40929

Description: If the (to be thought of as disjoint, although the proof does not require this) union of a set and its Hartogs number supports a group structure (more generally, a cancellative magma), then the set must be numerable. (Contributed by Stefan O'Rear, 9-Jul-2015.)

Assertion

Ref	Expression
isnumbasgrplem2	⊢ ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) → 𝑆 ∈ dom card)

Proof of Theorem isnumbasgrplem2

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		basfn 16916	. . 3 ⊢ Base Fn V
2		ssv 3945	. . 3 ⊢ Grp ⊆ V
3		fvelimab 6841	. . 3 ⊢ ((Base Fn V ∧ Grp ⊆ V) → ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) ↔ ∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))))
4	1, 2, 3	mp2an 689	. 2 ⊢ ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) ↔ ∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
5		harcl 9318	. . . . . 6 ⊢ (har‘𝑆) ∈ On
6		onenon 9707	. . . . . 6 ⊢ ((har‘𝑆) ∈ On → (har‘𝑆) ∈ dom card)
7	5, 6	ax-mp 5	. . . . 5 ⊢ (har‘𝑆) ∈ dom card
8		xpnum 9709	. . . . 5 ⊢ (((har‘𝑆) ∈ dom card ∧ (har‘𝑆) ∈ dom card) → ((har‘𝑆) × (har‘𝑆)) ∈ dom card)
9	7, 7, 8	mp2an 689	. . . 4 ⊢ ((har‘𝑆) × (har‘𝑆)) ∈ dom card
10		ssun1 4106	. . . . . . . 8 ⊢ 𝑆 ⊆ (𝑆 ∪ (har‘𝑆))
11		simpr 485	. . . . . . . 8 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
12	10, 11	sseqtrrid 3974	. . . . . . 7 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ⊆ (Base‘𝑥))
13		fvex 6787	. . . . . . . 8 ⊢ (Base‘𝑥) ∈ V
14	13	ssex 5245	. . . . . . 7 ⊢ (𝑆 ⊆ (Base‘𝑥) → 𝑆 ∈ V)
15	12, 14	syl 17	. . . . . 6 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ∈ V)
16	7	a1i 11	. . . . . 6 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (har‘𝑆) ∈ dom card)
17		simp1l 1196	. . . . . . . 8 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑥 ∈ Grp)
18	12	3ad2ant1 1132	. . . . . . . . 9 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑆 ⊆ (Base‘𝑥))
19		simp2 1136	. . . . . . . . 9 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑎 ∈ 𝑆)
20	18, 19	sseldd 3922	. . . . . . . 8 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑎 ∈ (Base‘𝑥))
21		ssun2 4107	. . . . . . . . . . 11 ⊢ (har‘𝑆) ⊆ (𝑆 ∪ (har‘𝑆))
22	21, 11	sseqtrrid 3974	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (har‘𝑆) ⊆ (Base‘𝑥))
23	22	3ad2ant1 1132	. . . . . . . . 9 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → (har‘𝑆) ⊆ (Base‘𝑥))
24		simp3 1137	. . . . . . . . 9 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑐 ∈ (har‘𝑆))
25	23, 24	sseldd 3922	. . . . . . . 8 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑐 ∈ (Base‘𝑥))
26		eqid 2738	. . . . . . . . 9 ⊢ (Base‘𝑥) = (Base‘𝑥)
27		eqid 2738	. . . . . . . . 9 ⊢ (+g‘𝑥) = (+g‘𝑥)
28	26, 27	grpcl 18585	. . . . . . . 8 ⊢ ((𝑥 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑥) ∧ 𝑐 ∈ (Base‘𝑥)) → (𝑎(+g‘𝑥)𝑐) ∈ (Base‘𝑥))
29	17, 20, 25, 28	syl3anc 1370	. . . . . . 7 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → (𝑎(+g‘𝑥)𝑐) ∈ (Base‘𝑥))
30		simp1r 1197	. . . . . . 7 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
31	29, 30	eleqtrd 2841	. . . . . 6 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → (𝑎(+g‘𝑥)𝑐) ∈ (𝑆 ∪ (har‘𝑆)))
32		simplll 772	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑥 ∈ Grp)
33	22	ad2antrr 723	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → (har‘𝑆) ⊆ (Base‘𝑥))
34		simprl 768	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑐 ∈ (har‘𝑆))
35	33, 34	sseldd 3922	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑐 ∈ (Base‘𝑥))
36		simprr 770	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑑 ∈ (har‘𝑆))
37	33, 36	sseldd 3922	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑑 ∈ (Base‘𝑥))
38	12	ad2antrr 723	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑆 ⊆ (Base‘𝑥))
39		simplr 766	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑎 ∈ 𝑆)
40	38, 39	sseldd 3922	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑎 ∈ (Base‘𝑥))
41	26, 27	grplcan 18637	. . . . . . 7 ⊢ ((𝑥 ∈ Grp ∧ (𝑐 ∈ (Base‘𝑥) ∧ 𝑑 ∈ (Base‘𝑥) ∧ 𝑎 ∈ (Base‘𝑥))) → ((𝑎(+g‘𝑥)𝑐) = (𝑎(+g‘𝑥)𝑑) ↔ 𝑐 = 𝑑))
42	32, 35, 37, 40, 41	syl13anc 1371	. . . . . 6 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → ((𝑎(+g‘𝑥)𝑐) = (𝑎(+g‘𝑥)𝑑) ↔ 𝑐 = 𝑑))
43		simplll 772	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑥 ∈ Grp)
44	12	ad2antrr 723	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑆 ⊆ (Base‘𝑥))
45		simprr 770	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑑 ∈ 𝑆)
46	44, 45	sseldd 3922	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑑 ∈ (Base‘𝑥))
47		simprl 768	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑎 ∈ 𝑆)
48	44, 47	sseldd 3922	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑎 ∈ (Base‘𝑥))
49	22	ad2antrr 723	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → (har‘𝑆) ⊆ (Base‘𝑥))
50		simplr 766	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑏 ∈ (har‘𝑆))
51	49, 50	sseldd 3922	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑏 ∈ (Base‘𝑥))
52	26, 27	grprcan 18613	. . . . . . 7 ⊢ ((𝑥 ∈ Grp ∧ (𝑑 ∈ (Base‘𝑥) ∧ 𝑎 ∈ (Base‘𝑥) ∧ 𝑏 ∈ (Base‘𝑥))) → ((𝑑(+g‘𝑥)𝑏) = (𝑎(+g‘𝑥)𝑏) ↔ 𝑑 = 𝑎))
53	43, 46, 48, 51, 52	syl13anc 1371	. . . . . 6 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → ((𝑑(+g‘𝑥)𝑏) = (𝑎(+g‘𝑥)𝑏) ↔ 𝑑 = 𝑎))
54		harndom 9321	. . . . . . 7 ⊢ ¬ (har‘𝑆) ≼ 𝑆
55	54	a1i 11	. . . . . 6 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → ¬ (har‘𝑆) ≼ 𝑆)
56	15, 16, 16, 31, 42, 53, 55	unxpwdom3 40920	. . . . 5 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ≼* ((har‘𝑆) × (har‘𝑆)))
57		wdomnumr 9820	. . . . . 6 ⊢ (((har‘𝑆) × (har‘𝑆)) ∈ dom card → (𝑆 ≼* ((har‘𝑆) × (har‘𝑆)) ↔ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆))))
58	9, 57	ax-mp 5	. . . . 5 ⊢ (𝑆 ≼* ((har‘𝑆) × (har‘𝑆)) ↔ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆)))
59	56, 58	sylib 217	. . . 4 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ≼ ((har‘𝑆) × (har‘𝑆)))
60		numdom 9794	. . . 4 ⊢ ((((har‘𝑆) × (har‘𝑆)) ∈ dom card ∧ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆))) → 𝑆 ∈ dom card)
61	9, 59, 60	sylancr 587	. . 3 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ∈ dom card)
62	61	rexlimiva 3210	. 2 ⊢ (∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)) → 𝑆 ∈ dom card)
63	4, 62	sylbi 216	1 ⊢ ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) → 𝑆 ∈ dom card)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  Vcvv 3432   ∪ cun 3885   ⊆ wss 3887   class class class wbr 5074   × cxp 5587  dom cdm 5589   “ cima 5592  Oncon0 6266   Fn wfn 6428  ‘cfv 6433  (class class class)co 7275   ≼ cdom 8731  harchar 9315   ≼^* cwdom 9323  cardccrd 9693  Basecbs 16912  +_gcplusg 16962  Grpcgrp 18577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-1cn 10929  ax-addcl 10931

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-omul 8302  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-oi 9269  df-har 9316  df-wdom 9324  df-card 9697  df-acn 9700  df-nn 11974  df-slot 16883  df-ndx 16895  df-base 16913  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-grp 18580  df-minusg 18581

This theorem is referenced by:  isnumbasabl  40931  isnumbasgrp  40932